Prob Sheet 1: First Order Ordinary Differential Equations Module F13YT2 1. Integration revision. Find y(x), given that dy dy ex (i) ; (ii) = = x2 sin x. x dx 3 + 6e dx 2. Name the type and find the general solution for each of the following first order equations: dy dy x2 dy (i) = ; (ii) + 3y = x + e−2x ; (iii) x = x cos(2x) − y; dx y dx dx y 2 + 2xy dy dy = = 0; (iv) ; (v) xy 2 − x + (x2 y + y) dx x2 dx dy (vi) (xey + 2y) + ey = 0; dx dy + 1 − y sin x + e2y = 0 (vii) (cos x + 2xe2y ) dx dy (viii) + y = 2 y3 dt 3. Show that each of the following equations has two distinct solutions. dy dy (ii) y (i) = y 1/2 ; y(0) = 0; = x; y(0) = 0. dx dx Explain, in each case, why the hypotheses of Picard’s Theorem are not satisfied. dy + y 2 = 1. Hence sketch the solutions satisfying dx (i) y(0) = 1.1; (ii) y(0) = 0; (iii) y(0) = −1.1. Check your answer using Maple. 4. Find all constant solutions of dy Consider the equation = y 2 − x. Determine the points in the xy-plane at which dx dy dy dy (i) = 0; (ii) > 0; and (iii) < 0. Hence sketch the solution satisfying y(1) = 1. dx dx dx Check your answer using Maple. 5. 6. Consider the following first order differential equation dy = y − x. dx (∗) (a) Determine the points in the xy-plane at which dy dy dy = 0; (ii) > 0; and (iii) < 0. Hence, sketch the direction field of (∗). (i) dx dx dx (b) Sketch the solution of (∗) satisfying the initial condition y(0) = 12 in the direction field you have made for (a) (there is no need to solve the initial value problem explicitly).